A variety of methods exists for estimating \(\mu \)  and the unknown change-points, for instance binary segmentation [1]} and its variants [2]}, [3]}, [4]}, [5]}, (penalized) cost methods such as optimal paritioning [6]}, [7]}, [8]}, [9]}, [10]}, multiscale methods [11]}, [12]} and Bayesian approaches [13]}, [14]}, among many others; for further detail see [15]}, [16]}, [17]} and references therein. The empirical and theoretical properties of these methods typically rely on selecting appropriate choices for tuning parameters. For instance, optimal partitioning [6]} requires selection of the number of change-points to estimate, and then determines the location of these to minimise the residual sum of squares. Given the correct number of change-points, the optimal partitioning estimate of the change-point locations is minimax rate-optimal and its \(L_2\) -error rate is optimal up to log-factors (see Theorem REF  in Section REF  and the following discussion as well as (REF )). Other approaches require different tuning parameters, and for some of these methods, theoretically motivated choices of those tuning parameters can be very successful in idealised settings where the joint distribution of the \(\varepsilon _i\)  is known. However, their performance can deteriorate when the error distribution is misspecified, as is likely to be the case in practice, particularly when the errors have heavy tails or are heteroscedastic.
